Local Controlability, Trajectory Geometric Local Infimuma, and Second-Order Conditions in Optimal Control

Avakov, E. R. , Magaril-Il'yaev, G. G.
Vladikavkaz Mathematical Journal 2026. Vol. 28. Issue 1.

Abstract:
In optimal control theory, questions related to necessary optimality conditions and the controllability of a control system are among the fundamental ones. In this paper, for a control system of ordinary differential equations, the concept of its local controllability with respect to an arbitrary continuous function and the concept of a trajectory of a geometric local infimum are defined. These concepts are dual to each other in the sense that either the control system is locally controllable with respect to a given function or this function is a trajectory of a geometric local infimum. The concept of a trajectory of a geometric local infimum generalizes the concept of a trajectory of a local infimum (previously introduced by the authors), and generalizes the classical concept of an optimal trajectory. A trajectory of a local infimum is a function such that the objective functional attains its minimum, but, generally speaking, it is not a feasible trajectory and it is a uniform limit of such trajectories. An optimal trajectory may not exist, but the existence of a local infimum trajectory is clearly sufficient for applications. The previously mentioned duality between the concepts of local controllability with respect to an arbitrary continuous function and the trajectory of a geometric local infimum was investigated by the authors in the case, where the necessary conditions for the trajectory of a local infimum were first-order. In this paper we focus on second-order necessary conditions. Note that first-order and second-order necessary conditions for the trajectory of a local infimum are improved the corresponding classical conditions (in partuicular, the Pontryagin maximum principle). Our goal is to show that the introduction of more general concepts (local controllability with respect to an arbitrary function, the trajectory of a geometric local infimum) allows to a unified approach of controllability and optimality in optimal control. Our examples also play an important role.

Keywords: trajectory of geometric local infimum, local controllability, optimal control, second-order conditions.

+ References